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Module-I (T-5hr P-0hr P-6hr)
Metric Spaces: Definition and examples, open and closed spheres,
Project-1: Write a report on applications of metric spaces.
Project-2: Show that the class denote a set of all bounded sequence of real numbers is a metric space.
Project-3: Show that the class of all sequences is a metric space.
Module-II (T-3hr P-0hr P-4hr)
Neighbourhoods, Interior points. Open set
Project-4: Write a report on relation between boundary points, limit points. And their applications.
Project-5(Flip class): Prove theorems on Neighbourhoods, Interior points
Module-III (T-5hr P-0hr P-6hr)
Closed set, boundary points, limit points and isolated points, closure of a set, dense sets.
Project-6: Write a report on difference between open set closed set and limit point , interior point with appropriate examples.
Project-7: Write the difference between boundary points, limit points and isolated points explain through examples.
Project-8 (Flip class): Prove theorems on closed set and dense sets.
Module-IV (T-4hr P-0hr P-6hr)
Separable metric spaces, sequences in metric space, convergent sequences, Cauchy sequences
Project-9: Write a report on applications of convergent sequences and Cauchy sequences.
Project-10: Every convergent sequence is bounded. Is the reverse true? Justify your answer with suitable example.
Project-11(Flip class): Prove theorems on convergent sequences and Cauchy sequences.
Module-V (T-4hr P-0hr P-4hr)
Complete metric space, subspace, Cantor’s theorem. Continuous functions,
Project- 12(Flip class): Prove theorems on Complete metric space, subspace.
Project-13(Flip class): : Prove that addition, subtraction, multiplication and division (with condition) of continuous functions are continuous.
Module-VI (T-4hr P-0hr P-6hr)
Uniform continuity, Connectedness, connected subsets of separated sets.
Project-14(Flip class): Prove theorems on Uniform continuity and Connectedness.
Project- 15: Write the difference between continuity and uniform continuity and explain through a proper example..
Project-16 (Flip class): Prove theorems on connected subsets of separated sets.
Module-VII (T-5hr P-0hr P-4hr)
Disconnected sets, contraction mappings, Banach fixed point theorem.
Project- 17(Flip class): Geometrical interpretation of connected and disconnected sets.
Project-18: Write some applications of Banach fixed point theorem.
Text Books:
Reference Books:
Introduction
PPT link: https://www.slideshare.net/NaliniSPatil/metric-space-151515844
Study Material: Introduction to Metric spaces with examples
youtube link: https://youtu.be/6CLl5xx5X-Y
Examples on Metric Spaces
Study Material: http://more examples
Youtube link: https://youtu.be/etP21xln8iQ
Project-1: Write a report on applications of metric spaces.
Theorems on Metric Spaces.
Study Material: Theorems on metric space
youtube link: https://youtu.be/Af03P1xVNSs?list=PLbMVogVj5nJSxFihV-ec4A3z_FOGPRCo-
Project-2: Show that the class denote a set of all bounded sequence of real numbers is a metric space.
Project-3: Show that the class of all sequences is a metric space.
Project-4: Write a report on relation between boundary points, limit points. And their applications
Project-5(Flip class): Prove theorems on Neighbourhoods, Interior points
Project-6: Write a report on difference between open set closed set and limit point , interior point with appropriate examples.
Project-7: Write the difference between boundary points, limit points and isolated points explain through examples
Project-8 (Flip class): Prove theorems on closed set and dense sets.
sequences in metric space
PPT link: https://www.slideshare.net/VishangSingh/analysis-sequences-and-bounded-sequences
convergent sequences
youtube link: https://youtu.be/DZT8o7Y08Mw
youtube link: https://youtu.be/FgRslqEYTJI
Project-9: Write a report on applications of convergent sequences and Cauchy sequences
Project-10: Every convergent sequence is bounded. Is the reverse true? Justify your answer with suitable example.
Project-11(Flip class): Prove theorems on convergent sequences and Cauchy sequences.
Project-12(Flip class): Prove theorems on Complete metric space, subspace.
Continuous functions
PPT link: https://www.slideshare.net/sumanmathews/continuous-functions
youtubelink : https://youtu.be/hXkQqCBLRp8
Project- 13: Prove that addition, subtraction, multiplication and division (with condition) of continuous functions are continuous.
Theorems on Continuous functions
PPT link: https://www.slideshare.net/VishveshJasani/continuity-of-a-function
Project-14(Flip class): Prove theorems on Uniform continuity and Connectedness.
Project- 15: Write the difference between continuity and uniform continuity and explain through a proper example
Project-16 (Flip class): Prove theorems on connected subsets of separated sets.
Project- 17(Flip class): Geometrical interpretation of connected and disconnected sets.
Project-18: Write some applications of Banach fixed point theorem.
Kali Prasad Rath received his M.Tech. in Computer Science and & M.Sc. in Mathematics from Berhampur University.This is his 10th year at Centurion University of Technology and Management and his 17th year in teaching profession. He is associated with qualitative study on theory of difference equation.
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